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MATH 412 PROBLEM SET 8 SOLUTIONS Reading MATH 412 PROBLEM SET 8 SOLUTIONS Reading: • Hungerford All of Chapter 7 and 8.1 Practice Problems: These are “easier” problems critical for your understanding, but not to be turned in. Be sure you can do these! You may see them or something very similar on Quizzes and Exams. • 7.1: 25, • 7.2: 14 • 7.3: 11, 12, 19 • 7.4: 29, 31, 40-44. • 7.5: 11, 31 • 8.1: 15 Problems due Friday. Write out carefully, staple pages for A and B together. Staple pages for C and D together separately. Write your name and section number on each. A. Let S1 be the subset of C consisting of complex number of absolute value 1; that is 1 S := fz 2 C j jzj = 1g: (1) Prove that S1 is a subgroup of C×. (2) Prove that the map x −y S1 ! SL ( ) x + iy 7! 2 R y x is an injective group homomorphism. 1 (3) Prove that S is isomorphic to SO2(R), the group of orthogonal matrices of determinant 1. Use this to give a geometric interpretation of the group S1 that explains why some call it the “continuous rotation group." (4) For every positive integer n, find an element of order n in S1. (5) Find an element of infinite order in S1. (6) For each positive integer n, find a subgroup of S1 of order n. Solution. (1) Clearly, since 1 2 S1, we see S1 is non-empty. Also, if z; w 2 S1, then zw 2 S1, since jzwj = jzjjwj = 1, so S1 is closed under multiplication. Finally, if z 2 S1, then since zz−1 = 1, we know jzjjz−1j = j1j = 1: This tells us that jz−1j = 1, so S−1 is closed under taking inverses. Thus S1 is a subgroup of C×. x −y a −b (2) Since (x + iy)(a + bi) = (xa − yb) + i(ya + xb) We need to check that = y x b a (xa − yb) −(ya + xb) . This is true. (ya + xb)(xa − yb) 1 MATH 412 PROBLEM SET 8 SOLUTIONS 2 1 (3) The map in (2) defines an isomorphism from S to the image in GL2(R). To see the image is in SO2(R), note simply that the columns are orthonormal and the determinant is 1 since x2 + y2 = 1 by virtue of being in S1. cos(2π=n) − sin(2π=n) (4) The matrix is order n, hence so is cos(2π=n) + i sin(2π=n) 2 S1. sin(2π=n) cos(2π=n) (5) For an element of infinite order, take an irrational multiple of 2π. (6) The subgroup generated by the order n element cos(2π=n) + i sin(2π=n) has order n. B. Consider the group GL2(Zp) of invertible 2 × 2 matrices with entries in the field Zp. This is the group of units in M2(Zp). a a b (1) Prove that for any non-zero column ; the matrix 2 M ( ) is a non-unit if and only if c c d 2 Zp a b there exists n 2 such that n = : Zp c d 1 (2) Compute the order of GL2(Zp): 2 (3) Show that the subset of diagonal matrices in GL2(Zp) is an abelian subgroup of order (p − 1) . 2 (4) Show that the upper triangular matrices in GL2(Zp) form a non-abelian subgroup of order p(p−1) . (5) Find a subgroup of order p − 1 in GL2(Zp). 2 (6) Show that the subgroup SL2(Zp) of matrices of determinant 1 has index p − 1 in GL2(Zp). (7) Prove that two matrices A; B in GL2(Zp) are in the same coset of SL2(Zp) in GL2(Zp) if and only if det A = det B. Solution. (1) A matrix is a unit if and only if it is invertible, which happens if and only if its columns are linearly independent (from 217). Since we are assuming the first column is non-zero, the matrix fails to be invertible if and only if the second column is a multiple of the first. (2) There are p2 − 1 choices for the first column, and p2 − p for the second, using (1). So 2 2 2 jGL2(Zp)j = (p − 1)(p − p) = p(p + 1)(p − 1) . (3) For the diagonal matrices, we need to have non-zero elements on the diagonal (or the deter- minant would be zero). There are (p − 1)2 of these. We check that they commute with each other by direct computation. (4) The upper triangular matrices again must have non-zero entries on the diagonal (since the 2 determinant is non-zero) but the upper right entry can be arbitrary. So jU2(Zp)j = (p − 1) p: 1 1 1 1 1 1 1 1 To see it is non-abelian, check that 6= : 0 2 0 1 0 1 0 2 a 0 (5) The invertible scalar matrices f j a 2 ×g are a subgroup of order p − 1. 0 a Zp × (6) There is a surjective homomorphism GL2(Zp) ! Zp sending each matrix to its determinant. ∼ Its kernel is the subgroup SL2(Zp). By the first isomorphism theorem, GL2(Zp)=SL2(Zp) = × × Zp . So the number of cosets equals the order of Zp ; which is p − 1. So the index is p − 1. (7) Suppose that A; B are in the same (right) coset of SL2(Zp). This means that A = CB where C 2 SL2(Zp): So det(A) = det(BC) = det(B) det(C) = det(B): Conversely, if det A = det B, −1 −1 then AB has determinant one, so is in SL2(Zp). Let C = AB . Then A = CB, where 1Hint: You might want to use (1) in the course of doing this. 2You can do this one after you do (7) if you like. MATH 412 PROBLEM SET 8 SOLUTIONS 3 C 2 SL2(Zp). This means that A and B are in the same (right) coset of SL2(Zp). [Note that SL2(Zp) is normal in GL2(Zp) so its left and right cosets are the same.] C. Consider the following elements in GL2(C): 1 0 i 0 0 1 0 i 1 = ; i = ; j = ; k = ; 0 1 0 −i −1 0 i 0 Let Q be the subgroup of GL2(C) generated by the matrices i; j; k: (1) Show3 that Q contains the 8 elements {±1; ±i; ±j; ±kg: (2) Make a multiplication table for Q. (3) Find (with proof) the complete list of all cyclic subgroups of Q of order 4. (4) Find (with proof) the complete list of all cyclic subgroups of Q of order 2. (5) Prove that every proper subgroup of Q is cyclic. (6) Can Q be generated by two elements? Prove it. (7) Is Q8 isomorphic to D4? Prove or disprove. Solution. (1) ◦ 1 −1 i −i j −j k −k 1 1 −1 i −i j −j k −k −1 −1 1 −i i −j j −k k i i −i −1 1 k −k −j j −i −i i 1 −1 −k k j −k j j −j −k k −1 1 i −i −j −j j k −k 1 −1 −i i k k −k j −j −i i −1 1 −k −k k −j j i −i 1 −1 (2) Identifying cyclic subgroups of order 4 amounts to finding elements of order 4. Note that ±i; ±j; ±k are all order 4, and −1 is order 2. So there are there cyclic groups of order 4, namely, those generated by i; j or k. Note that hii = fi; −1; −i; 1g, and similarly for j and k. (3) There is exactly one subgroup of order 2, namely {±1g. (4) The proper subgroups are order 1, 2, and 4 by Lagrange’s theorem. We have listed all the non-trivial ones in the preceding two problems. All are cyclic. (5) Yes, Q = hi; ji. Note that if we have i and j, we have ij = k, and together the powers of i; j; andk yield the rest of the elements of Q. (6) No. D4 has order eight, but it has five element of order 2 (the four reflecttions and rotation through 180). The group Q has one element of order 2. D. A regular n-gon is a planar figure with n vertices and n congruent sides, with all interior angles congruent. For example, a regular 3-gon is another word for an equilateral triangle and a regular 4-gon is the same as a square. Let Dn be the symmetry group of a regular n-gon. To fix ideas, fix a regular n-gon in the Cartesian plane so that its vertices are equidistant from the origin and one lies on the x-axis. It is a fact that jDnj = 2n. You may use this without proof for this problem (though it may be instructive to think about why this is so). 3It might be helpful to do (2) more or less at the same time. MATH 412 PROBLEM SET 8 SOLUTIONS 4 (1) The group Dn contains an element r of order n. Describe it. (2) For each vertex v of the regular n-gon, Dn contains an of order 2. Describe it. (3) Show that for n odd, the reflections in (2) give n different elements of Dn but if n is even, they give only n=2 different elements of Dn. (4) If n is even, find n=2 reflections in Dn not already described in (3).
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